\newproblem{lay:5_3_32}{
  % Problem identification
	\begin{large}
	  \hspace{\fill}\newline
    \textbf{Lay, 5.3.32}
	\end{large}
	\\
  \ifthenelse{\boolean{identifyAuthor}}{\textit{Carlos Oscar Sorzano, Aug. 31st, 2013} \\}{}

  % Problem statement
	Construct a $2\times 2$ matrix that is diagonalizable but not invertible.
}{
   % Solution
	Consider $P=\begin{pmatrix}1 & 0 \\ 0 & 1\end{pmatrix}$ and $D=\begin{pmatrix}1 & 0 \\ 0 & 0\end{pmatrix}$. Now let us construct the matrix
	\begin{center}
		$A=PDP^{-1}=\begin{pmatrix}1 & 0 \\ 0 & 0\end{pmatrix}$
	\end{center}
	It is obviously diagonalizable by construction, but it is not invertible because one of its eigenvalues is 0, and $D$ is not invertible.
}
\useproblem{lay:5_3_32}
\ifthenelse{\boolean{eachProblemInOnePage}}{\newpage}{}
